Questions involving philosophy of mathematics

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Computability, Continuity and Constructivism

Triggered by an IMO extremely interesting question & Mathematics Stack Exchange, asked by Dal: Computability and continuous real functions And a link in one of the comments that could have ...
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Every planar graph can be embedded on a sphere - formal proof?

The proof of the following theorem: A graph can be embedded on the surface of a sphere without crossings if and only if it can be embedded in the plane without crossings. is very short- The ...
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Given ∼R and ∼ B, derive ∼ (R ∨ B)

*This question deals with the derivation system SD (Sentential Derivation), the rules of which can be seen on pages 3-4 here: http://www.shamik.net/teaching/materials/dasgupta%20SL%20definitions.pdf ...
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Why is homeomorphism understood as stretching and bending?

A function $f: X \to Y$ between two topological spaces $(X, T_X)$ and $(Y, T_Y)$ is called a homeomorphism if it has the following properties: $f$ is a bijection (one-to-one and onto) $f$ is ...
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How do people who study intensely abstract mathematics “imagine” or understand the concepts they are studying or being taught? [closed]

This question is probably to the actual people who study such mathematics, rather than any "third-party". I haven't studied any such mathematics, but I can imagine that some (probably most of it) of ...
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The word “times” for multiplication…? [closed]

The word "times" for multiplication operation which is quite touching to the concept of time (feeling time this way 0*1=0). When was introduced that term? Did any other language have the kind of term ...
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Question about induction to infinity with regard to Bolzano's philosophy

I'm a philosophy and mathematics student, and I'm writing a paper on a proof put forward by Bolzano that if we can know one thing to be true, then we can know infinite truths. Put simply, he states ...
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Induction in a minimalistic foundation

Below the non-infinite axioms of Zermelo, with informal explanations. $(1)\quad\forall x \forall y ( \forall z(z \in x \leftrightarrow z \in y) \rightarrow x = y)$ Extensionality: $A=B\iff(x\in ...
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Why is mathematical induction a valid proof technique?

Context: I'm studying for my discrete mathematics exam and I keep running into this question that I've failed to solve. The question is as follows. Problem: The main form for normal induction over ...
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Resource request: do you think mathematics was invented or discovered? [closed]

I am doing an EPQ on this topic and any resources such as books or websites relating to this would be great. Any supported opinions would also be appreciated.
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Why does randomness exhibit a pattern in the long run?

!!! Layman here so please avoid complex math and answers. Random (usually pseudorandom) events are usually characterized along these lines: Each outcome in a trial experiment must be i.i.d.; i.e. ...
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What is the difference between a counter-intuitive statement and a paradox?

In mathematics and logic, what is the difference between a counter-intuitive statement and a paradox? For example, what differs something like the Banach-Tarski theorem or Gabriel's horn from ...
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What's the difference between “unprovable” and “undecidable”?

It seems to me that there is a difference between an unprovable sentence, and an undecidable sentence, but sometimes I have the impression that some authors use the terms interchangeably. In my ...
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Sheaves in Philosophy

I once found a book on google.books. It was about the applications of sheave theory to philosophy or more general to social studies. I don't remember for sure. i just know it was not the book Sheaves ...
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What does if-then has to do with not being true?

I'm reading Chihara's: Constructibility and Mathematical Existence. It says: An even more radical view rejects the assumption that mathematics is true—at least in the straightforward way that ...
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Does taking courses in mathematics give any help for mathematical logic?

I'm undergraduate student of philosophy department and I think I'll major in mathematical logic. For studying mathematical logic, I thought studying math lectures would give help to logic. So I ...
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Can mathematics get from other sciences what it got from physics?

Throughout history, physics has been an unparalleled source of '' inspiration'' for discovering/inventing mathematical ideas, which is due to its ability to describe the physical world. But can this ...
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Is mathematics invented or discovered? [closed]

In physics for example, and in science in general, facts are "discovered" in the sense that they arise from observing nature. A particle is discovered if we can measure its existence in nature. A law ...
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Are there logical arguments against modern $\sf ZFC$ set theory?

As of asking this question, my knowledge of set theory is quite pedestrian. I've read about it in numerous nontechnical papers and even worked through three chapters of Jech - Set Theory, but in terms ...
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Is the anti-foundation axiom considered constructive?

In the area of theoretical computer science that I am interested in, constructive mathematics is of practical interest because it gives algorithms that can be implemented on a computer. However, ...
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philosophy : first axiom of geometry and variable curvature

The very first axiom of geometry can be described as: Two different points lay on one and only one line. And I was wondering are there surfaces where this axiom irrecoverably fails? and I found ...
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Why not both true and false?

Why can't some mathematical statement (or whatever is the correct term) be both true and false? For example we can prove (e.g. by induction) that $1+2+3+\cdots+n=\frac{n(n+1)}{2}$ for all positive ...
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What's behind the Banach-Tarski paradox? [closed]

The discovery of the Banach-Tarski paradox was of course a great thing in mathematics but raises the issue of the relation between mathematics and reality. Empirically there are good reasons for faith ...
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Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
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Question on induction technique

When one uses induction (say on $n$) to prove something, does it mean the proof holds for all finite values of $n$ or does it always hold when even $n$ takes $\pm\infty$?
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Why do we first introduce the open set definition for continuity instead of the neighborhood definition?

After (nearly) completing my course in topology, something weird just stuck out to me which I hadn't considered before. When first discussing continuity, we often use the following definition: Let ...
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Properties of the simplest object in n-dimension

In my boredom, I was thinking about why the simplest 3d object (i.e. the one with the least faces, sides, vertices) was the tetrahedron. After it made sense to me, I realized some cool stuff which was ...
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Why do we focus so much in math on functions (as a subclass of relations)?

Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and ...
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Are there undecidable problems for which a solution has been found?

I mean are there examples of problems that have been proven to be undecidable, in the sense that it would not be possible to devise a deterministic computer program that outputs a solution for an ...
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Is ( Set S contains $x$ and only $x$ then does S equals $x$ ) true?

If a set $S$ contains $x$ and only $x$, then does $S$ equal $x$?
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1answer
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Goldbach's conjecture can't be proved to be undecidable?

Conjectures concerning natural numbers which could be settled by a counterexample can, as far as I understand, not be proved to be undecidable without being proved not having a counterexample at the ...
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the purpose of induction

After getting an answer (in a comment) from peter for this question I have a follow up question. If, in all horses are the same color problem for example, we need to use reason, reason which is ...
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How much of Mathematics is limited by our writing? [closed]

I'm sorry if this question is too vague or otherwise a stupid question. Suppose the mathematicians in some alien civilisation similar to ours sculpted their Mathematics in three dimensions (or ...
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Is there a name for this constant? (0.0100011011…)

It's the simplest number I could think of that contains any finite binary code in its digits: $$\begin{align} c &= 0.0100011011000001010011100101110111...\\ &= ...
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Apparent Arbitrariness in Mathematics

Something about definitions in mathematics has always interested – confused? - me, I call it “arbitrariness in Mathematics” - it's a bad name, but I don't know a better one. Let me explain: 1st - ...
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maximum number of possible rules of a sequence?

Read this http://www.mathsisfun.com/algebra/sequences-finding-rule.html and also http://en.wikipedia.org/wiki/The_Oxford_Murders_%28film%29 where the scene about the murder note left behind contains a ...
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A Question Regarding Representing $\mathscr P$($\omega$) as a Digraph and CH

It is well known that one can represent sets as digraphs. What is the proper digraph representation of $\mathscr P$($\omega$)? I ask this because $\mathscr P$($\omega$) is $\Pi_1$ in the Levy ...
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Impredicativity and set theory

I have thought about an example in set theory, but I don't know if its legal to do it, maybe someone can help. Let $\emptyset$ be given and let $A$ be a non empty set. Let us create the subset $X = ...
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What is the “opposite” of the Axiom of Choice?

One might think that, trivially, the "opposite" of AC is $\neg$AC. However, thinking about it differently, I'm not sure this is intuitively the case. AC says that every set has a choice function. ...
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Philosophical implications of P vs NP proof?

Wikipedia article on P vs NP says that "a proof either way would have profound implications for ... Philosophy" without providing further details. So I was wondering what could be the philosophical ...
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Why is the Power Set Operation Inherently Vague?

It is a somewhat common view among mathematicians/philosophers (who have an opinion on the subject) that the power set operation is inherently vague. They go on to say that its inherent vagueness is ...
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What goes wrong in the following argument that our conception of “set” is inconsistent?

This question might sound facetious, but it is a genuine question which I am very much interested in. I apologize in advance if it is too conceptual or philosophical, but I'm optimistic that I might ...
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Can arithmetic truths fix the truth value of the Continuum Hypothesis?

Many logicians and philosophers believe that all sentences expressible in the language of Peano Arithmetic have determinate truth-values, even though no nice formal system can capture all of these ...
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Is the first-order incompleteness of a theory (like arithmetics, set theory or logic itself) avoidable in a second or higher-order axiomatizations?

Can we avoid the first-order incompleteness of a theory (like arithmetics or set theory) in a second-order theory which contains the previous? How does it depend on the chosen semantics or models? If ...
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Strange Consequences of Large Cardinals in Probability

Large cardinal axioms are very strong hypothesizes and as any other strong hypothesis they have many strange consequences in mathematics. On the other hand we know that if we bring even the least ...
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Is Bell's Notion of “Abstract Set” Flawed?

Consider the following definition of "abstract set" given by John L. Bell (who wrote the book "Set Theory: Boolean-valued Models and Independence Proofs") from his preprint "Abstract and Variable ...
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Why the dual of some results are true while others are false?

In mathematics, many results have their "dual" versions. In many cases, if a result is true, then its dual is true as well. However, there are some examples while the dual of a true statement is ...
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Good Sources for Lecture Movies in Set Theory, Logic and Philosophy of Maths

Of course as any other researcher I'm not able to attend any scientific event in my research area. But it is always interesting and useful to watch the lecture movies of these events. I will ...
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what is $e$ really? what is its meaning? [duplicate]

I don't get it how we came up with $e$ and how can nature use this number so much! that is what I have been told and I only know that $e$ is a specific constant like $\pi$! I understand that $\pi$ ...
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Can a proof be too long? [closed]

Suppose that there is an omnipotent oracle saying that there is a proof for Riemann hypothesis but its proof is so long the universum would collapse before mankind will understand the proof. Would ...